Consider the optimization problem:
Minimize Trace(X) subject to
| ATX + XA + XBBTX + Q < 0 | (1) |
with data
It can be shown that the minimizer X* is simply the stabilizing solution of the algebraic Riccati equation
ATX + XA + XBBTX + Q = 0
This solution can be computed directly with the Riccati solver care
and compared to the minimizer returned by mincx.
From an LMI optimization standpoint, the problem specified in Equation 1 is equivalent to the following linear objective minimization problem:
Minimize Trace(X) subject to
| (2) |
Since Trace(X) is a linear function of the entries of
X, this problem falls within the scope of the mincx solver and can be numerically solved as follows:
Define the LMI constraint of Equation 1 by the sequence of commands
setlmis([]) X = lmivar(1,[3 1]) % variable X, full symmetric lmiterm([1 1 1 X],1,a,'s') lmiterm([1 1 1 0],q) lmiterm([1 2 2 0],-1) lmiterm([1 2 1 X],b',1) LMIs = getlmis
Write the objective Trace(X) as cTx where x is the vector of free entries of X. Since c should select the diagonal entries of X, it is obtained as the decision vector corresponding to X = I, that is,
c = mat2dec(LMIs,eye(3))
Note that the function defcx provides a more
systematic way of specifying such objectives (see Specifying cTx Objectives for mincx for
details).
Call mincx to compute the minimizer
xopt and the global minimum copt =
c'*xopt of the objective:
options = [1e-5,0,0,0,0] [copt,xopt] = mincx(LMIs,c,options)
Here 1e–5 specifies the desired relative accuracy on
copt.
The following trace of the iterative optimization performed by mincx appears on the
screen:
Solver for linear objective minimization under LMI constraints Iterations : Best objective value so far
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Result: feasible solution of required accuracy best objective value: -18.716695 guaranteed relative accuracy: 9.50e-06 f-radius saturation: 0.000% of R = 1.00e+09
The iteration number and the best value of cTx at the current iteration appear in the left and right columns, respectively. Note that no value is displayed at the first iteration, which means that a feasible x satisfying the constraint (Equation 2) was found only at the second iteration. Lower bounds on the global minimum of cTx are sometimes detected as the optimization progresses. These lower bounds are reported by the message
*** new lower bound: xxx
Upon termination, mincx reports that the global
minimum for the objective Trace(X) is –18.716695 with
relative accuracy of at least 9.5×10–6. This is
the value copt returned by mincx.
mincx also returns the
optimizing vector of decision variables xopt. The
corresponding optimal value of the matrix variable X is
given by
Xopt = dec2mat(LMIs,xopt,X)
which returns
This result can be compared with the stabilizing Riccati solution computed by
care:
Xst = care(a,b,q,-1) norm(Xopt-Xst) ans = 6.5390e-05